Consistent estimation of a mean planar curve modulo similarities

We consider the problem of estimating a mean planar curve from a set of $J$ random planar curves observed on a $k$-points deterministic design. We study the consistency of a smoothed Procrustean mean curve when the observations obey a deformable model including some nuisance parameters such as random translations, rotations and scaling. The main contribution of the paper is to analyze the influence of the dimension $k$ of the data and of the number $J$ of observed configurations on the convergence of the smoothed Procrustean estimator to the mean curve of the model. Some numerical experiments illustrate these results.